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We characterize the outcome of majority voting for single-peaked preferences on median spaces. This large class of preferences covers a variety of multi-dimensional policy spaces including products of lines (e.g. grids), trees, and hypercubes. Our main result is the following: If a Condorcet winner (i.e. a winner in pairwise majority voting) exists, then it coincides with the appropriately defined median ("the median voter"). This result generalizes previous findings which are either restricteddoi:10.1007/s00355-013-0744-0 fatcat:h5leofbmqrdwrfhqrj6bcdqvta